We also point out to the interested reader that a first version of this article
was considering a variant of the model of level-\(k\)  phylogenetic networks,
where multiple (i.e. parallel) edges are allowed.
The counting results for this alternate model of course differ (starting from level 2),
and can be found in [1]}, again with an associated Maple documentavailable at http://user.math.uzh.ch/bouvel/publications/BouvelGambetteMansouri_Version1_WithMultipleEdges.mw.
Similarly, these files can easily be used to adapt the computations in case other restrictions are imposed on the structure of level-1 or level-2 phylogenetic networks, for example if tiny cycles, defined in [2]} as bridgeless components with exactly three vertices, are not allowed.
